The radial wavefunction $g(r)$ for the hydrogen atom satisfies the equation

$$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d g(r)}{d r}\right)-\frac{e^{2} g(r)}{4 \pi \epsilon_{0} r}+\hbar^{2} \frac{\ell(\ell+1)}{2 m r^{2}} g(r)=E g(r)$$

With reference to the general form for the time-independent Schrödinger equation, explain the origin of each term. What are the allowed values of $\ell$ ?

The lowest-energy bound-state solution of $(*)$, for given $\ell$, has the form $r^{\alpha} e^{-\beta r}$. Find $\alpha$ and $\beta$ and the corresponding energy $E$ in terms of $\ell$.

A hydrogen atom makes a transition between two such states corresponding to $\ell+1$ and $\ell$. What is the frequency of the emitted photon?